The abstract group concept

Version for printingThis article is based on a lecture given by Peter Neumann (a son of Bernhard Neumann and Hanna Neumann) at a conference at the University of Sussex on 19 March 2001 to celebrate the 90th birthday of Walter Ledermann. The talk was entitled Introduction to the theory of finite groups the title of the famous text written by Walter Ledermann. This article is based on notes taken by EFR at that lecture.

The modern definition of a group is usually given in the following way.

Definition
A groupG is a set with a binary operation G × G → G which assigns to every ordered pair of elements x, y of G a unique third element of G (usually called the product of x and y) denoted by xy such that the following four properties are satisfied:

Closure: if x, y are in G then xy is in G.

Associative law: if x, y, z are in G then x(yz) = (xy)z.

Identity element: there is an element e in G with ex = xe = x for all x in G.

Inverses: for every x in G there is an element u in G with xu = ux = e.

The first point to make is that 1. is debatable as an axiom since it is a consequence of the definition of a binary operation. However it is not our purpose to debate this here and it is convention that this axiom is included.

Where did this, now standard, definition come from? Particularly we wish to examine some moves towards this definition made in the 19th century. We are not, therefore, concerned here with the bulk of the work done in group theory in the 19th century which concerned the study of permutation groups required for Galois theory. It is important to realise that the abstract definition of a group was merely an esoteric sideline of group theory through the 19th century.

Let us first note that there were two meanings of the term "abstract group" during the first half of the 20th century - say from 1905 to 1955. One meaning was that of a group defined by the four axioms as above, while the second meaning was that of a group defined by generators and relations. For example Todd used the term in this second way when he talked about "the Mathieu groups as abstract groups". Here we are only interested in the first of these meanings.

The emergence of the abstract group concept was a remarkably slow process. The story begins with the prehistory which involves Galois and Cauchy. Galois defined a group in 1832 although it did not appear in print until Liouville published Galois' papers in 1846. The first version of Galois' important paper on the algebraic solution of equations was submitted to the Paris Académie des Sciences in 1829. René Taton has found evidence in the archives of the Académie which suggest that Cauchy spoke with Galois and persuaded him to withdraw the paper and submit a new version of it for the Grand Prix of 1830. This is based on strong circumstantial evidence, but we do know that Galois submitted to Fourier a new version of his paper to be considered for the Grand Prix in March 1830.

Fourier died shortly after this and Galois' paper was lost. It was never considered for the prize which was awarded jointly to Abel (posthumously) and Jacobi in July 1830. Galois was invited by Poisson to submit a third version of his memoir on equations to the Académie and he did so on 17 January 1831. This version of the paper was refereed by Poisson who rejected it but wrote a very sympathetic report. Although Galois had proved the results in general, the paper only considered equations of prime degree. Poisson failed to understand the paper and suggested that the arguments were developed further. It was unclear to him how Galois' results classified which equations were soluble by radicals.

The night before he fought the duel which led to his death Galois made notes on his papers. One of these notes, made on 29 May 1832, was the following:-

... if in such a group one has the substitutions S and T then one has the substitution ST.

Although Galois had used groups extensively throughout his paper on equations, he had not given a definition. It is little wonder that Poisson found the paper hard to understand for it contains many explicit calculations in a group, yet the concept was not defined - poor Poisson!

Now in 1845, one year before Liouville published the above definition by Galois, Cauchy gave a definition. He considered substitutions in n letters x, y, z, ... and defined derived substitutions to be all those which can be deduced by multiplying these substitutions together in any order. He then called the set of substitutions together with the derived substitutions, a "conjugate system of substitutions". For some time these two identical concepts, a "group" and a "conjugate system of substitutions", were both used. However, from 1863 when Jordan wrote a commentary on Galois' work in which he used "group", it became the standard term. This was reinforced when Jordan published his major group theory text Traité des substitutions et des équations algebraique in 1870. However, Cauchy's term "conjugate system of substitutions" continued to be used by some up to about 1880.

How much was Cauchy influenced by Galois? Although he had seen Galois' papers submitted to the Académie, there was no explicit definition of a group in them. On the other hand he must have at least been subliminally influenced. Both Galois and Cauchy, of course, define groups in terms of the closure property alone. The now familiar axioms of associativity, identity and inverses do not appear. Both were dealing with permutations which means that closure is all that is necessary to define a group, the other properties all following automatically. Cauchy went on to write 25 papers on this topic between September 1845 and January 1846.

[Note by EFR. Peter Neumann did not mention in his lecture any influence that Ruffini's ideas might have had on Cauchy. In 1821 Cauchy had written to Ruffini praising his work which he had clearly read. Although there is no explicit definition of a group in Ruffini's work, again the concept clearly appears and may have had as major an influence on Cauchy's thinking as the work of Galois.]

The first person to try to give an abstract definition of a group was Cayley. He wrote a paper on groups in 1854 which he published again in two separate journals in 1878. In the 1854 paper he attempted to give an abstract definition in terms of symbols θ which operate on a system (x, y, ... ) so that

Cayley went on to define an identity symbol 1 which leaves the members of the system unaltered. He defines the element θφ as the result of operating on the system first by φ then by θ. He notes that θφ need not equal φθ. Cayley also requires the associative law θ.φψ = θφ.ψ to be satisfied. He then says that any set of such symbols with the property that the product of any two is in the set is called a group.

This is an important attempt at an abstract definition of a group but Cayley has overreached himself. What he has here is really a mess. Why does he require the associative law to hold if his symbols are operators? As for permutations the associative law follows automatically for operators. It is also not clear that (x', y', ... ) where x', y', ... are any functions of x, y, ... will be in the system. This definition is not entirely successful.

This idea was picked up by Burnside, von Dyck and others. Burnside, in his book The Theory of Groups of Finite Order published in 1897 gave the following definition:-

Let A, B, C, ... represent a set of operations which can be performed on the same object or set of objects.

He then supposes that any two of the operations are distinct in the sense that no two produce the same effect. He follows Cayley in requiring closure, the associative law, and inverses. Again his definition is subject to the same criticism at Cayley's definition. If his elements are operations then why does he need to postulate the associative law? Rather strangely Burnside does not assume the existence of an identity, although one can infer it from the fact that A and A-1 are in the set and we have closure. Burnside repeats exactly the same definition in the second edition of his book which appeared in 1911.

It is worth noting that neither Cayley nor Burnside insisted that their groups were finite. The definition deliberately allowed for the possibility of infinite groups, and Burnside in particular was interested in studying infinite groups. What we have given here is part of a sequence of development which we might call the English school. There were other bits in the sequence between these major contributions which we have omitted. We now turn to another development of group theory which was going on at the same time which we might call the European school.

In 1870 Kronecker gave a definition of a group in a completely different context, namely the context of a class group in algebraic number theory. He takes a specifically finite set θ', θ'', θ''', ... such that from any two, a third can be derived by a specific method. He then assumes that the commutative and associative laws hold and also that θ'θ'' ≠ θ'θ''' if θ'' ≠ θ'''. This appears to be a separate development by Kronecker who does not tie it in with previous work on groups. However, Heinrich Weber in 1882 gave a very similar definition to that of Kronecker yet he did tie it in with previous work on groups.

Heinrich Weber defined a group of degree h, like Kronecker in the context of class groups, again to be a finite set. He required that from two elements of the system one can derive a third element of the system so that the following hold:-

(θrθs)θt = θr(θsθt) = θrθsθt

θθr = θθsimpliesθr = θs.

A few comments here. The associative law is put in a slightly strange way. The expression θrθsθt has no meaning until the associative law is defined to hold, so in a modern treatment one might write:

(θrθs)θt = θr(θsθt) so that either side may be denoted by θrθsθt.

One can still see that there is a little lack of clarity in the definition. What Heinrich Weber is in fact defining is a semigroup with cancellation and, given that it is finite, this is sufficient to ensure the existence of an identity and of inverses. That this fails for infinite systems was noted by Heinrich Weber in his famous text Lehrbuch der Algebra published in 1895. In this book he notes that the above definition of a group only works for finite groups and in the infinite case one needs to postulate inverses explicitly.

Here is a diagram to illustrate the interactions between the various mathematicians working their way slowly towards the abstract definition of a group.

We should point out that there appears to be relatively little cross-fertilisation between the two lines of development. There was some, however. For instance Burnside did read Frobenius's papers, although he did so later than he should have done and had to apologise twice to Frobenius for not knowing what he had already published. Recently a letter from Burnside to Schur has been discovered and we know that Burnside corresponded with Hölder.

We do know where some of the 20th century influences came from - Emmy Noether, an important 20th century figure particularly influential through van der Waerden's Algebra book, was strongly influenced by Heinrich Weber. On the other hand Schur, another influential 20th century figure, was influenced by Frobenius.

Some anecdotes about Ledermann's book, Introduction to the theory of finite groups are at this link